The Ultimate Calculus Course

1. Differential Calculus: This area deals with the concept of the derivative, which allows us to understand rates of change. For instance, if you’re looking at how a car accelerates, differential calculus helps you determine the car’s speed at any given moment. It involves understanding slopes of curves and how these slopes change.
2. Integral Calculus: This area focuses on the concept of the integral, which is essentially about accumulation. For example, if you know the rate at which a tank is being filled with water, integral calculus lets you figure out how much water is in the tank over time. It involves adding up infinitely many tiny quantities to find the whole.

Both of these concepts rely on the fundamental notion of limits, which is the idea of approaching a value (but not necessarily reaching it). Calculus has vast applications in science, engineering, economics, statistics, and many other fields. It provides a framework for modeling systems in which there is change, and a way to predict and understand the behavior of those systems.

The Absolute Best Book to Ace Calculus

Calculus for Beginners The Ultimate Step by Step Guide to Acing Calculus

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Calculus Complete Course

Functions

• Function Notation
• Adding and Subtracting Functions
• Multiplying and Dividing Functions
• Composition of Functions
• Writing Functions
• Graphing Functions
• Parent Functions
• Function Inverses
• Inverse Variation
• Domain and Range of function
• Piecewise Function
• Positive, Negative, Increasing and Decreasing Functions on Intervals

Advanced Functions

• Exponential Function
• Linear, Quadratic and Exponential Models
• Linear vs Exponential Growth
• Logarithms
• Properties of logarithms
• Natural Logarithms
• Sine, Cosine, and Tangent
• Reciprocal Functions: Cosecant, Secant, and Cotangent
• Domain and Range of Trigonometric Functions
• Trigonometric Function Values for Key Angles
• The Unit Circle
• Additional trigonometric reminders
• Periodic properties of trigonometric functions
• Floor and Ceiling Functions

Sequences and Series

• Arithmetic Sequences
• Geometric Sequences
• Sigma Notation (Summation Notation)
• Arithmetic Series
• Geometric Series
• Binomial Theorem
• Pascal’s Triangle
• Alternate Series

Limit and Continuity

• Limit Introduction
• Neighborhood
• Estimating Limits from Tables
• Functions with Undefined Limits (from table)
• Functions with Undefined Limits (from graphs)
• One Sided Limits
• Limit at Infinity
• Continuity at a Point
• Continuity over an Interval
• Removing Discontinuity
• Direct Substitution
• Limit Laws
• Limit Laws Combinations
• The Squeeze Theorem
• Indeterminate and Undefined
• Infinity cases
• Trigonometric Limits
• Rationalizing Trigonometric Functions
• Algebraic Manipulation
• Redefining function’s value
• Rationalizing Infinite Limits

Derivative

• Derivative introduction
• Average and instantaneous rates of change
• The derivative of a function
• Rules of differentiation
• Derivative of trigonometric functions
• Power rule
• Product rule
• Quotient rule
• Chain rule
• Power rule combined with other derivative rules
• Derivative of radicals
• Derivative of logarithms and exponential functions
• L’Hôpital
• Differentiability
• Second derivatives: Minimum vs. Maximum
• Curve sketching using derivatives
• Differentiating Inverse Functions
• Optimization problems
• Implicit differentiation
• Related rates

Integrals

• What is Integral?
• Applications of Integrals
• Exponential Growth and Decay 
• The Anti-Derivative
• Riemann Sums
• Rules of Integration
• Power Rule (of Integration)
• Fundamental Theorem of Calculus
• Trigonometric Integrals
• Substitution Rule
• Integration by Parts
• Integral of Radicals
• Exponential and Logarithmic Integrals
• Improper Integrals

Differential Equations

• Introduction and applications
• Classification of Differential Equations
• First-Order Ordinary Differential Equations
• Linear Differential Equations
• Separable Differential Equations
• Slope Fields
• Euler’s Method for Numerical Solutions
• Simple Growth and Decay
• Population Models

Analytic Geometry

• Ellipses, parabolas, and hyperbolas
• Polar Coordinates
• Converting Between Polar and Rectangular Coordinates
• Graphing Polar Equations
• Applications of Polar Coordinates

Complex Numbers

• Complex Numbers addition and subtraction
• Multiplying and Dividing Complex Numbers
• Rationalizing Imaginary Denominators

The Best Books for High School Students

The Ultimate High School Math Bundle From Algebra I to Pre-Calculus

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Recommended EffortlessMath Books

For a workbook that pairs with this course, the Calculus for Beginners (resource currently unavailable) walks through every topic with worked examples. For AP-specific prep, see the AP Calculus AB Test Prep Bundle (resource currently unavailable) and the AP Calculus BC Test Prep Bundle (resource currently unavailable).

Frequently Asked Questions

What is a calculus course?

A calculus course teaches the math of change. A first course, often called Calculus 1 or single-variable calculus, covers limits, derivatives and their uses, such as optimization and related rates, then integrals and the Fundamental Theorem of Calculus, which links the two. For example, if you know a car’s position over time, the derivative gives its speed at each moment.

Is calculus class hard?

Calculus feels hard to many students, but usually because of shaky algebra and trigonometry rather than the new ideas themselves. If you can factor, work with exponents and logarithms, and use the unit circle comfortably, derivatives and integrals become much more manageable. Steady practice helps most: one lesson and a short worksheet each day beats cramming before a test.

What is the AP Calculus course?

AP Calculus is a College Board course taken in high school that can earn college credit with a qualifying exam score. AP Calculus AB covers limits, derivatives, integrals and their applications. AP Calculus BC includes everything in AB plus topics such as infinite series, parametric and polar functions, and more integration techniques. Credit policies differ by college, so check with the schools you are considering.

What are the calculus courses in order?

At most colleges the sequence is Calculus 1 (limits and derivatives, often ending with basic integrals), Calculus 2 (integration techniques, applications of integrals, sequences and series) and Calculus 3 (multivariable calculus with partial derivatives, multiple integrals and vectors). Differential equations and linear algebra often follow. Before Calculus 1, students usually finish Algebra 2 and Pre-Calculus. Course content varies, so check your school’s catalog.

What is the hardest calculus course?

Many students say Calculus 2 is the hardest, because it combines many integration techniques and introduces infinite series, which feel different from earlier topics. Others find Calculus 3 tougher because it asks you to picture surfaces and vectors in three dimensions. Which one challenges you most depends on your strengths, but solid algebra and trigonometry make every level easier.

Is there a free calculus course?

Yes. The Ultimate Calculus Course on Effortless Math is free and self-paced. It covers single-variable calculus from limits through derivatives, the chain rule, optimization, related rates, integrals, areas and volumes. Each topic has a lesson with worked examples and a free worksheet with worked-out solutions, which suits high school AP Calculus students and first-year college students.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

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